In this work we study the problem of {\em Stochastic Budgeted Multi-round Submodular Maximization} (SBMSm), in which we would like to adaptively maximize the sum over multiple rounds of the value of a monotone and submodular objective function defined on a subset of items, subject to the fact that the values of this function depend on the realization of stochastic events and the number of items that we can select over all rounds is limited by a given budget. This problem extends, and generalizes to multiple round settings, well-studied problems such as (adaptive) influence maximization and stochastic probing. We first show that, if the number of items and stochastic events is somehow bounded, there is a polynomial time dynamic programming algorithm for SBMSm. Then, we provide a simple greedy approximation algorithm for SBMSm, that first non-adaptively allocates the budget to be spent at each round, and then greedily and adaptively maximizes the objective function by using the budget assigned at each round. Such algorithm guarantees a $(1-1/e-\epsilon)$-approximation to the optimal adaptive value. Finally, by introducing a metric called {\em budget-adaptivity gap}, we measure how much an optimal policy for SBMSm, that is adaptive in both the budget allocation and item selection, is better than an optimal partially adaptive policy that, as in our greedy algorithm, determined the budget allocation in advance. We show a tight bound of $e/(e-1)$ on the budget-adaptivity gap, and this result implies that our greedy algorithm guarantees the best approximation among all partially adaptive policies.

翻译：本文研究随机预算多轮次模最大化问题，其目标是在多轮次中自适应地最大化定义在物品子集上的单调模函数值之和，该函数值取决于随机事件的实现，且所有轮次中可选择的物品总数受给定预算限制。该问题将影响最大化与随机探测等经典研究推广至多轮次场景。首先证明当物品数量与随机事件规模有界时，存在多项式时间动态规划算法求解该问题。随后提出一种简易贪心近似算法：该算法首轮非自适应地分配各轮预算，随后在各轮使用分配预算通过贪心策略自适应地最大化目标函数。该算法可保证达到最优自适应值的(1-1/e-ε)近似比。最后通过引入预算自适应间隙度量指标，量化了在预算分配与物品选择均自适应的最优策略相较于预算分配预先确定的局部自适应策略（如本文贪心算法）的优越性。研究证明预算自适应间隙存在紧界e/(e-1)，该结果表明本文贪心算法在所有局部自适应策略中具有最优近似保证。